Morse homology

In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories.

Given any (compact) smooth manifold, let f be a Morse function and g a Riemannian metric on the manifold. (These are auxiliary; in the end, the Morse homology depends on neither.) The pair (f, g) gives us a gradient vector field. We say that (f, g) is Morse–Smale if the stable and unstable manifolds associated to all of the critical points of f intersect each other transversely.

For any such (f, g), it can be shown that the difference in index between any two critical points is equal to the dimension of the moduli space of gradient flows between those points. Thus there is a one-dimensional moduli space of flows between a critical point of index i and one of index i − 1. Each flow can be reparametrized by a one-dimensional translation in the domain. After modding out by these reparametrizations, the quotient space is zero-dimensional — that is, a collection of oriented points representing unparametrized flow lines.

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